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May 17th, 2017, 01:35 AM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 230 Thanks: 2  Differentiable function
Let $f: R > R $ be a twice differentiable function. Then which of the following is true ? A) If $f(0)=0= f''(0) $ then $f'(0)=0$. B) $f$ is a polynomial. C) $f'$ is continuous. D) If $f''(x) > 0$ for all $x$ in $R$ then $f(x)>0$ for all $x$ in $R$. I have checked some examples for the option A and it is satisfying this condition. So I guess the answer is A. Please let me know if I'm wrong 
May 17th, 2017, 03:25 AM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,737 Thanks: 1387 
Does choice A hold true for $f(x)=\sin{x}$ ?

May 17th, 2017, 03:52 AM  #3 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 230 Thanks: 2  
May 17th, 2017, 03:56 AM  #4 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 230 Thanks: 2  
May 17th, 2017, 04:57 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,274 Thanks: 2435 Math Focus: Mainly analysis and algebra 
Yes. A necessary condition for the derivative of a function to exist (at a point) is that the function be continuous (at that point). If we $f$ is twice differentiable, then $f'$ is differentiable and thus continuous. Almost all functions we look at (especially in learning differentiation) are infinitely (also known as "continuously") differentiable. But not all functions are. The classic example is $$f(x)=\begin{cases} x^2\sin\frac1x &(x \ne 0) \\ 0 & (x=0)\end{cases}$$ which is differentiable only once with $f'$ being disc9ntinuous at $x=0$. 

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